Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

4 Issues per year


IMPACT FACTOR 2017: 0.658

CiteScore 2017: 1.05

SCImago Journal Rank (SJR) 2017: 1.291
Source Normalized Impact per Paper (SNIP) 2017: 0.893

Mathematical Citation Quotient (MCQ) 2017: 0.76

Online
ISSN
1609-9389
See all formats and pricing
More options …
Volume 18, Issue 4

Issues

A Hybrid Method for Solving Inhomogeneous Elliptic PDEs Based on Fokas Method

Eleftherios-Nektarios G. Grylonakis
  • Department of Electrical and Computer Engineering, School of Engineering, Democritus University of Thrace, University Campus, Kimmeria, GR 67100 Xanthi, Greece
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Christos K. Filelis-Papadopoulos
  • Department of Electrical and Computer Engineering, School of Engineering, Democritus University of Thrace, University Campus, Kimmeria, GR 67100 Xanthi, Greece
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ George A. Gravvanis
  • Corresponding author
  • Department of Electrical and Computer Engineering, School of Engineering, Democritus University of Thrace, University Campus, Kimmeria, GR 67100 Xanthi, Greece
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-12-05 | DOI: https://doi.org/10.1515/cmam-2017-0053

Abstract

In this paper we propose a hybrid method for solving inhomogeneous elliptic PDEs based on the unified transform. This approach relies on the derivation of the global relation, containing certain integral transforms of the given boundary data as well as of the unknown boundary values. Herewith, the approximate global relation for the Poisson equation is solved numerically using a collocation method on the complex λ-plane, based on Legendre expansions. The corresponding numerical results are presented using closed-form expressions and numerical approximations for different types of boundary and source data, indicating the applicability of the considered approach. Additionally, the full solution is computed in a recursive manner by splitting the domain into smaller concentric polygons, and by using a spatial-stepping scheme followed by an interpolation step. Furthermore, numerical results are also given for the solution of the Poisson and the inhomogeneous Helmholtz equations on several convex polygons. Additional results are provided for the case of nonconvex polygons as well as for the case of a problem with discontinuities across an interface. The proposed approach provides a framework for solving inhomogeneous elliptic PDEs using the unified transform.

Keywords: Poisson Equation; Helmholtz Equation; Unified Transform; Global Relation; Collocation, Interpolation

MSC 2010: 65N22; 65N35; 65R10

References

  • [1]

    A. C. L. Ashton and A. S. Fokas, Elliptic equations with low regularity boundary data via the unified method, Complex Var. Elliptic Equ. 60 (2015), no. 5, 596–619. Web of ScienceCrossrefGoogle Scholar

  • [2]

    C. F. Chan Man, D. De Kee and P. N. Kaloni, Advanced Mathematics for Engineering and Science, World Scientific, Hackensack,, 2003. Google Scholar

  • [3]

    C.-I. R. Davis and B. Fornberg, A spectrally accurate numerical implementation of the Fokas transform method for Helmholtz-type PDEs, Complex Var. Elliptic Equ. 59 (2014), no. 4, 564–577. CrossrefWeb of ScienceGoogle Scholar

  • [4]

    A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 453 (1997), no. 1962, 1411–1443. CrossrefGoogle Scholar

  • [5]

    A. S. Fokas, Two-dimensional linear partial differential equations in a convex polygon, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2006, 371–393. CrossrefGoogle Scholar

  • [6]

    A. S. Fokas, A new transform method for evolution partial differential equations, IMA J. Appl. Math. 67 (2002), no. 6, 559–590. CrossrefGoogle Scholar

  • [7]

    A. S. Fokas, Boundary-value problems for linear PDEs with variable coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), no. 2044, 1131–1151. CrossrefGoogle Scholar

  • [8]

    A. S. Fokas, A Unified Approach to Boundary Value Problems, CBMS-NSF Regional Conf. Ser. in Appl. Math. 78, Society for Industrial and Applied Mathematics, Philadelphia, 2008. Google Scholar

  • [9]

    A. S. Fokas, A. Iserles and S. A. Smitheman, The unified transform in polygonal domains via the explicit Fourier transform of Legendre polynomials, Unified Transform for Boundary Value Problems, Society for Industrial and Applied Mathematics, Philadelphia (2015), 163–171. Google Scholar

  • [10]

    B. Fornberg and N. Flyer, A numerical implementation of Fokas boundary integral approach: Laplace’s equation on a polygonal domain, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), no. 2134, 2983–3003. CrossrefGoogle Scholar

  • [11]

    S. R. Fulton, A. S. Fokas and C. A. Xenophontos, An analytical method for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 167 (2004), no. 2, 465–483. CrossrefGoogle Scholar

  • [12]

    E.-N. G. Grylonakis, C. K. Filelis-Papadopoulos and G. A. Gravvanis, A note on solving the generalized Dirichlet to Neumann map on irregular polygons using generic factored approximate sparse inverses, CMES Comput. Model. Eng. Sci. 109 (2015), no. 6, 505–517. Google Scholar

  • [13]

    P. Hashemzadeh, A. S. Fokas and S. A. Smitheman, A numerical technique for linear elliptic partial differential equations in polygonal domains, Proc. A. 471 (2015), no. 2175, Article ID 20140747. Web of ScienceGoogle Scholar

  • [14]

    A. S. Kronrod, Nodes and Weights of Quadrature Formulas. Sixteen-Place Tables, Consultants Bureau, New York, 1965. Google Scholar

  • [15]

    A. N. Marques, J.-C. Nave and R. R. Rosales, A correction function method for Poisson problems with interface jump conditions, J. Comput. Phys. 230 (2011), no. 20, 7567–7597. Web of ScienceCrossrefGoogle Scholar

  • [16]

    L. F. Shampine, MATLAB program for quadrature in 2D, Appl. Math. Comput. 202 (2008), no. 1, 266–274. Web of ScienceGoogle Scholar

  • [17]

    A. G. Sifalakis, A. S. Fokas, S. R. Fulton and Y. G. Saridakis, The generalized Dirichlet–Neumann map for linear elliptic PDEs and its numerical implementation, J. Comput. Appl. Math. 219 (2008), no. 1, 9–34. CrossrefWeb of ScienceGoogle Scholar

  • [18]

    E. A. Spence and A. S. Fokas, A new transform method II: The global relation and boundary-value problems in polar coordinates, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466 (2010), no. 2120, 2283–2307. CrossrefGoogle Scholar

  • [19]

    E. Süli and D. F. Mayers, An Introduction to Numerical Analysis, Cambridge University Press, Cambridge, 2003. Google Scholar

  • [20]

    Z. Yosibash, Numerical analysis on singular solutions of the Poisson equation in two-dimensions, Comput. Mech. 20 (1997), no. 4, 320–330. CrossrefGoogle Scholar

About the article

Received: 2017-04-23

Revised: 2017-08-02

Accepted: 2017-10-23

Published Online: 2017-12-05

Published in Print: 2018-10-01


Citation Information: Computational Methods in Applied Mathematics, Volume 18, Issue 4, Pages 653–672, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2017-0053.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in